Bifurcation,chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.     

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

We consider the dynamics of a harmonically forced oscillator with an asymmetric elastic–perfectly plastic stiffness function. The computed bifurcation diagrams for the oscillator show regions of periodic motion, hysteresis and large regions of chaotic motion. These different regions of dynamical behaviour are plotted in a two-dimensional parameter space consisting of forcing amplitude and forcing frequency. Examples of the chaotic motion encountered are shown using a discontinuity crossing map. Comparisons are made with the symmetric oscillator by computing a typical bifurcation diagram and considering previously published results for the symmetric system. From this we conclude that the asymmetric system is dominated by a large region of chaotic motion whereas in the symmetric oscillator period one motion and coexisting period three motion predominates.     

Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback

The chaotic behavior of a double-well Duffing oscillator with both delayed displacement and velocity feedbacks under a harmonic excitation is investigated. By means of the Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. The analytical results reveal that for negative feedback the presence of time delay lowers the threshold and enlarges the possible chaotic domain in parameter space; while for positive feedback the presence of time delay enhances the threshold and reduces the possible chaotic domain in parameter space, which are further verified numerically through Poincare maps of the original system. Furthermore, the effect of the control gain parameters on the chaotic motion of the original system is studied in detail.     

Analysis of effect of random perturbation on dynamic response of gear transmission system

In order to investigate the effects of random perturbation of a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness, the random dynamic model of a single pair of three-degree-of-freedom spur gear transmission system is established. With gear meshing frequency changing, the dynamic characteristics of the gear transmission system were analyzed by bifurcation diagram, phase diagram, time course diagram and Poincaré map of the system. The effects of random perturbation caused by a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness were comparative analyzed. Numerical simulation shows that the gear transmission system with nonlinear clearance exists rich period-doubling bifurcation phenomenon. With the increasing of the gear meshing frequency, gear transmission system will be from the chaotic motion to periodic motion by inverse period-doubling bifurcation. The effect of the meshing frequency random perturbation on the gear transmission system movement is largest. On the contrary, the effect of the meshing stiffness random perturbation on the system is minimum.     

Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension

This study performs a systematic analysis of the dynamic behavior of a single degree-of-freedom spur gear system with and without nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a spur gear system and therefore serve as a useful source of reference for engineers in designing and controlling such systems.     

Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos

This study aims to analyze the dynamic behavior of bevel-geared rotor system supported on a thrust bearing and journal bearings under nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

Strong nonlinearity analysis for gear-bearing system under nonlinear suspension—bifurcation and chaos

This study performs a systematic analysis of the dynamic behavior of a gear-bearing system with nonlinear suspension, nonlinear oil-film force, and nonlinear gear mesh force. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

Chaotic responses on gear pair system equipped with journal bearings under turbulent flow

This study performs a systematic analysis of the dynamic behavior of a gear-bearing system with the turbulent flow effect, nonlinear suspension, nonlinear oil-film force and nonlinear gear mesh force. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient, unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic and chaotic behaviors. The results presented in this study provide a detailed understanding of the operating conditions under which undesirable dynamic motion takes place in gear-bearing system and therefore offer a useful source of reference for engineers in designing and controlling such systems.     

Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection

This paper presents the conditions that can possibly lead to chaotic motion and bifurcation behavior for a simply-supported large deflection thermo-elastic circular plate with variable thickness by utilizing the criteria of fractal dimensions, maximum Lyapunov exponents and bifurcation diagrams. The governing partial differential equation of the simply supported thermo-elastic circular plate with variable thickness is first derived by means of Galerkin method. Several different features including Fourier spectra, phase plot, Poincar’e map and bifurcation diagrams are numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitations of lateral loads and thermal loads. Numerical examples are presented to verify the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. Numerical modeling results indicate that large deflection motion of a thermo-elastic circular plate with variable thickness possesses chaotic motions and bifurcation motion under different lateral loads and thermal loads. The simulation results also indicate that the periodic motion of a circular plate can be obtained for the convex or the concave circular plate. The dynamic motion of the circular plate is periodic for the cases including (1) the lateral loading frequency is within a specific range, (2) thermal and lateral loadings are operated in a specific range and (3) the thickness parameter is less than a specific critical value for the convex circular plate or greater than a specific critical value for the concave circular plate. The modeling results show that the proposed method can be employed to predict the non-linear dynamics of any large deflection circular plate with variable thickness.     

Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays

Time delays are often sources of complex behavior in dynamic systems. Yet its complexity needs to be further explored, particularly when multiple time delays are present. As a purpose to gain insight into such complexity under multiple time delays, we investigate the mechanism for the action of multiple time delays on a particular non-autonomous system in this paper. The original mathematical model under consideration is a Duffing oscillator with harmonic excitation. A delayed system is obtained by adding delayed feedbacks to the original system. Two time delays are involved in such system, one of which in the displacement feedback and the other in the velocity feedback. The time delays are taken as adjustable parameters to study their effects on the dynamics of the system. Firstly, the stability of the trivial equilibrium of the linearized system is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or double Hopf bifurcation may occur. Then, the chaotic behavior of such system is investigated in detail. Particular emphasis is laid on the effect of delay difference between two time delays on the chaotic properties. A Melnikov’s analysis is employed to obtain the necessary condition for onset of chaos resulting from homoclinic bifurcation. And numerical analyses via the bifurcation diagram and the top Lyapunov exponent are carried out to show the actual time delay effect. Both the results obtained by the two analyses show that the delay difference between two time delays plays a very important role in inducing or suppressing chaos, so that it can be taken as a simple but efficient “switch” to control the motion of a system: either from order to chaos or from chaos to order.     

Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.     

Global chaos in a periodically forced, linear system with a dead-zone restoring force

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are introduced through switching planes pertaining to two constraints. The global periodic motions based on the Poincare mapping are determined, and the eigenvalue analysis for the stability and bifurcation of periodic motion is carried out. Global chaos in such a system is investigated numerically from the unstable global periodic motions analytically determined. The bifurcation scenario with varying parameters is presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed in this investigation.     

Homoclinic bifurcation and chaos control in MEMS resonators

The chaotic dynamics of a micromechanical resonator with electrostatic forces on both sides are investigated. Using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters. Detailed numerical studies including basin of attraction, and bifurcation diagram confirm the analytical prediction and reveal the effect of parametric excitation amplitude on the system transition to chaos. The main result of this paper indicates that it is possible to reduce the electrostatically induced homoclinic and heteroclinic chaos for a range of values of the amplitude and the frequency of the parametric excitation. Different active controllers are applied to suppress the vibration of the micromechanical resonator system. Moreover, a time-varying stiffness is introduced to control the chaotic motion of the considered system. The techniques of phase portraits, time history, and Poincare maps are applied to analyze the periodic and chaotic motions.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

Nonlinear responses of a rub-impact overhung rotor

For a rotor system with bearings and step-diameter shaft in the oxygen pump of an engine, the contact between the rotor and the case is considered, and the chaotic response and bifurcation are investigated. The system is divided into elements of elastic support, shaft and disk, and based on the transfer matrix method, the motion equation of the system is derived, and solved by Newmark integration method. It is found that hardening the support can delay the occurrence of chaos. When rubbing begins, the grazing bifurcation will cause periodic motion to become quasi-period. With variation of system parameters, such as rotating speed, imbalance and external damping, chaotic response can be observed, along with other complex dynamics such as period- doubling bifurcation and torus bifurcation in the response.     

Response and bifurcation of rotor with squeeze film damper on elastic support

This paper investigates the nonlinear response and bifurcation of rotor with Squeezed Film Damper (SFD) supported on elastic foundation. The motion equations are derived. To analyze the bifurcation of nonlinear response of SFD rotor, the Floquet Multipliers is obtained by solving the perturbation equations with numerical method. For computing Floquet Multipliers, a novel method is presented in this paper, which can begin integration at the stable solution. Simulation results are given in two figures. One figure, which consists of eight subfigures, gives the effect of rotating speed on the response of SFD damper supported on elastic foundation: with increasing rotating speed, the nonlinear response evolves from quasi-period to period, then jumps between different periods, and finally returns to quasi-period; the corresponding bifurcations are saddle-node bifurcation and secondary Hopf bifurcation. The second figure, which consists of six subfigures, shows that: the support stiffness has large influence on the response of bearings and film force in SFD; large support stiffness can lead to oil whirl in SFD.     

Dynamic complexities in a parasitoid-host-parasitoid ecological model

Chaotic dynamics have been observed in a wide range of population models. In this study, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented. The model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well. Many forms of complex dynamics were observed, including pitchfork bifurcation with quasi-periodicity, period-doubling cascade, chaotic crisis, chaotic bands with narrow or wide periodic window, intermittent chaos, and supertransient behavior. Furthermore, computation of the largest Lyapunov exponent demonstrated the chaotic dynamic behavior of the model.     

Chaos in the unbalance response of a rigid rotor in cavitated squeeze-film dampers without centering springs

Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.     

Chaotic and bifurcation for a simply-supported thermo-mechanical coupling circular plate with variable thickness

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads , η₁=1.1, β=0.5, and _∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection.     

Nonlinear dynamics of unicycles in leader–follower formation

In this paper, a dynamical systems analysis is presented for characterizing the motion of a group of unicycles in leader–follower formation. The equilibrium formations are characterized along with their local stability analysis. It is demonstrated that with the variation in control gain, the collective dynamics might undergo Andronov–Hopf and Fold–Hopf bifurcations. The vigor of quasi-periodicity in the regime of Andronov–Hopf bifurcation and heteroclinic bursts between quasi-periodic and chaotic behavior in the regime of Fold–Hopf bifurcation increases with the number of unicycles. Numerical simulations also suggest the occurrence of global bifurcations involving the destruction of heteroclinic orbit.